Results 1 to 2 of 2

Math Help - Question invovling cyclinderical volume and such...

  1. #1
    Zell_49
    Guest

    Question invovling cyclinderical volume and such...

    Ok. I am stumped in my calculus class and need help with one of the problems. I am supposed to find the dimensions (r and h) of the cylinder that will minimize the cost of materials, knowing that V is constant, if the cost of the top and bottom are .15 per square inch and the side is .07 per square inch. (Note that my answer will be in terms of V.). help?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Apr 2005
    Posts
    1,631
    "Ok. I am stumped in my calculus class and need help with one of the problems. I am supposed to find the dimensions (r and h) of the cylinder that will minimize the cost of materials, knowing that V is constant, if the cost of the top and bottom are .15 per square inch and the side is .07 per square inch. (Note that my answer will be in terms of V.). help?"

    Umm, I didn't see this question before.

    One way to solve this is to express the Cost C in terms of a related variable, say radius r, and then set dC/dr = 0

    Cost C = 2(pi r^2)*(0.15) +(2pi r h)(0.07)
    C = (0.3pi)(r^2) +(0.14pi)(r)(h) -----------(i)
    We try to eliminate the h so that C will be in terms of r only.

    Volume of cylinder, V = (pi r^2)(h)
    So, h = V / (pi r^2) ---------------------(ii)
    Substitute that into (i),
    C = (0.3pi)(r^2) +(0.14pi)(r)(V / pi r^2)
    C = (0.3pi)(r^2) +(0.14V)/r ------------(iii)
    There. There is only one variable for C. Remember that V is constant as mentioned.

    Differentiate both sides of (iii) with respect to r,
    dC/dr = (0.6pi)(r) +(0.14V)(-1/ r^2)
    dC/dr = (0.6pi)(r) -(0.14V)/(r^2)
    Set dC/dr to zero,
    0 = (0.6pi)(r) -(0.14V)/(r^2)
    (0.14V)/(r^2) = (0.6pi)(r)
    Clear the fraction, multiply both sides by r^2,
    0.14V = (0.6pi)(r^3)
    r^3 = (0.14V)/(0.6pi)
    r = cuberoot(0.14V / 0.6pi)
    r = (0.42)*[V^(1/3)] inches ----------answer.

    Substitute that into (ii),
    h = V / (pi r^2) ---------------------(ii)
    h = V / [pi (0.42 V^(1/3))^2]
    h = V / [pi (0.1764 V^(2/3)]
    h = [V^(1 -2/3)] / (0.1764pi)
    h = (1.8)*[V^(1/3)] inches ----------------answer
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. probability inequality invovling function
    Posted in the Advanced Statistics Forum
    Replies: 4
    Last Post: January 27th 2011, 01:46 AM
  2. Volume question
    Posted in the Calculus Forum
    Replies: 1
    Last Post: July 28th 2010, 02:28 PM
  3. word problem invovling rate change
    Posted in the Calculus Forum
    Replies: 3
    Last Post: March 12th 2010, 08:07 AM
  4. Volume question
    Posted in the Calculus Forum
    Replies: 1
    Last Post: March 4th 2010, 04:22 AM
  5. Replies: 1
    Last Post: March 17th 2008, 11:28 PM

/mathhelpforum @mathhelpforum